<!DOCTYPE html>
<html lang="en-US">
<!--********************************************-->
<!--*       Generated from PreTeXt source      *-->
<!--*                                          *-->
<!--*         https://pretextbook.org          *-->
<!--*                                          *-->
<!--********************************************-->
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<meta name="robots" content="noindex, nofollow">
</head>
<body class="ignore-math">
<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Given a function <span class="process-math">\(f(x)\)</span> defined on <span class="process-math">\([-L, L]\text{,}\)</span> one can always take integrations to write out a Fourier series <span class="process-math">\(\hat{f}(x)\text{.}\)</span> The problem is if we can use <span class="process-math">\(\hat{f}(x)\)</span> to approximate <span class="process-math">\(f(x)\)</span> on <span class="process-math">\([-L, L]\text{?}\)</span> Or simply, does <span class="process-math">\(\hat{f}(x)\)</span> converges to <span class="process-math">\(f(x)\)</span> on <span class="process-math">\([-L, L]\text{?}\)</span> Is it possible for a series in <a href="" class="xref" data-knowl="./knowl/fourier.html" title="Equation 7.3.1">(7.3.1)</a> to converge at a point <span class="process-math">\(x\)</span> in the interval <span class="process-math">\((-L, L)\text{,}\)</span> yet not be equal to <span class="process-math">\(f(x)\text{?}\)</span> The answer is Yes.</p>
<span class="incontext"><a href="sec7_4.html#p-352" class="internal">in-context</a></span>
</body>
</html>
